# Tagged Questions

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### Mobius transformations are bijections proof

I don't understand the last line of this proof. To show a function is bijective we need to show it is one-to-one and onto. The proof shows that $f$ is one-to-one only. For some reason $f^{-1}$ ...
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### Sequence of Mobius Transformation

Let $T(z) = \frac {z+2}{2z+1}$. Now it follows that: $T_1(z) = T(z), T_2(z) = T(T_1(z)), T_3(z)=T(T_2(z)) .... T_{n+1}(z)=T(T_n(z))$ I'm trying to prove this sequence at the nth terms, but I ...
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### Möbius transformation image

Let $f(z)=\frac{az+b}{z+d}$, when $d\in\mathbb{R}$, $d\not=0$ $a,b\in\mathbb{C}$ and $f$ is not constant. I want to find the image of the real and imaginary axes under $f$. I've found that the image ...
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### Query regarding Linear Transformation…

As we always read in Complex Analysis, Linear Transformation (L.T.) is a combination of Translation, Rotation and Magnification i.e. $T(z)=az+b$ is a L.T. in complex. However, It doesn't satisfy the ...
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### Transformations on the complex plane

I'm trying to work out what the transformation $T:z \rightarrow -\frac{1}{z}$ does (eg reflection in a line, rotation around a point etc). Any help on how to do this would be greatly appreciated! I've ...
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### Complex Variables Conformal Mapping in Complex Plane of harmonic Functions

Consider the harmonic function $u(x,y) = 1 - y + x/(x^2+y^2)$ on the upper half plane $y > 0$. What is the corresponding harmonic function on the first quadrant $x>0$, $y>0$, under the ...
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### Real linear tranformation

When do we say that a transformation $T$ which takes the complex number field onto itself is real-linear? I need to know it for my homework but I can't seem to find the definition anywhere.
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### The image under mapping $w=(z+i)/(z-i)$, of the third quadrant?

The title says it all. I am not sure how to approach this problem. The only related problems i have done is mapping a (unbounded)line /circle to a line/circle. Regards Exatic
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### Mapping behavior of imaginary axis via $v=\frac{z-a}{z+a}$

I would like to know what the bilinear transform $v=\frac{z-a}{z+a}$ does to the imaginary axis, where $a$ is a real number. I substituted $z=yi$ and calculated $|v|$ giving me $|v| =1$. Is this ...
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### inverse transform of $Z(\omega) =\frac{a}{\alpha-i\omega}$

I am stuck at calculating the inverse transorm of $Z(\omega) =\frac{a}{\alpha-i\omega}$. Can someone help me please? thanks
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### A Möbius transformation maps circles and lines to circles and lines. What exactly does that mean?

The title pretty much says it all. I am also looking for a concrete example if possible. I have looked at the proof, but I'm not exactly sure what it means because I am kind of confused on what the ...
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### Möbius transformation question

Möbius transformation copies the annulus $\{z:r<|z|<1\}$ to the domain between $\{z:|z-1/4|=1/4\}$ and $\{z:|z|=1\}$ Please help me to find what is $r$.
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### What is the image of $D=\{z:0<\operatorname{Re}z<\pi\}\setminus\{\pi/2\}$ under $f(z)=\tan z$?

What would be the image of the domain $D = \{z:0<\operatorname{Re}z<\pi\} \setminus \{\pi/2\}$ under $f(z) = \tan z$? I havn't met with tan(z) transformation so I don't really know how to ...
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### Harmonic Function Transformation Help

Consider the harmonic function $$u(x,y)=1-y+\frac{x}{x^2+y^2}$$ on the upper half plane $y>0$. What is the corresponding harmonic function on the first quadrant $x>0$, $y>0$, under the ...
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